Similar to ordinal models, the framework for the binary probit model given by
equation (REF ) can be utilized to describe a binary logit
model (or simply logit model) with the modification that the errors follow a
logistic distribution [1]}. Both the location and scale
restrictions still apply to the logit model, but note that the variance is
now fixed at \(\pi ^{2}/3\)  as compared to 1 in a probit model. To obtain the
logit likelihood, the normal cdf \(\Phi (x^{\prime }_{i}\beta )\)  in
equation (REF ) is replaced by the logistic cdf:
\(\Lambda (x^{\prime }_{i}\beta ) = \exp {(x^{\prime }_{i}\beta )}/ \big [1 + \exp {(x^{\prime }_{i}\beta )}\big ] \) . We can maximize the resulting log-likelihood to obtain the parameter
estimates for the logit model.
